Rong Kong

Efficient, automated Monte Carlo methods for radiation transport

Monte Carlo simulations provide an indispensible model for solving radiative transport problems, but their slow convergence inhibits their use as an everyday computational tool. In this paper, we present two new ideas for accelerating the convergence of Monte Carlo algorithms based upon an efficient algorithm that couples simulations of forward and adjoint transport equations. Forward random walks are first processed in stages, each using a fixed sample size, and information from stage k is used to alter the sampling and weighting procedure in stage k + 1. This produces rapid geometric convergence and accounts for dramatic gains in the efficiency of the forward computation. In case still greater accuracy is required in the forward solution, information from an adjoint simulation can be added to extend the geometric learning of the forward solution. The resulting new approach should find widespread use when fast, accurate simulations of the transport equation are needed.

Sequential Correlated Sampling Methods for Some Transport Problems

In this paper, we will describe how to extend to the solution of certain simple particle transport problems a sequential correlated sampling method introduced by Halton in 1962 for the efficient solution of certain matrix problems. Although the methods apply quite generally, we have so far studied in detail only problems involving planar geometry.

A new proof of geometric convergence for general transport problems based on sequential correlated sampling methods

In [J. Halton, Sequential Monte Carlo, Proc. Comb. Phil. Soc. 58 (1962), J. Halton, Sequential Monte Carlo Techniques for the Solution of Linear Systems, J. Sci. Comp. 9 (1994) 213-257] Halton introduced a strategy to be used in Monte Carlo algorithms for the efficient solution of certain matrix problems. We showed in [R. Kong, J. Spanier, Sequential correlated sampling methods for some transport problems, in: Harold Niederreiter, Jerome Spanier (Eds.), Monte-Carlo and Quasi Monte-Carlo Methods 1998: Proceedings of a Conference at the Claremont Graduate University, Springer-Verlag, New York, 2000, R. Kong, J. Spanier, Error analysis of sequential Monte Carlo methods for transport problems, in: Harold Niederreiter, Jerome Spanier (Eds.), Monte-Carlo and Quasi Monte-Carlo Methods 1998: Proceedings of a Conference at the Claremont Graduate University, Springer-Verlag, New York, 2000] how Haltons method based on correlated sampling can be extended to continuous transport problems and established their geometric convergence for a family of transport problems in slab geometry. In our algorithm, random walks are processed in batches, called stages, each stage producing a small correction that is added to the approximate solution developed from the previous stages. In this paper, we demonstrate that strict error reduction from stage to stage can be assured under rather general conditions and we illustrate this rapid convergence numerically for a simple family of two dimensional transport problems.

Error Analysis of Sequential Monte Carlo Methods for Transport Problems

In 1962, Halton introduced a sequential correlated sampling algorithm for the efficient solution of certain matrix problems. We have extended Halton’s method to the solution of certain simple transport problems and the resulting algorithm is capable of producing geometric convergence for these problems. In our algorithm, random walks are processed in groups, called stages, and the result of each stage is a small correction that is added to the solution at the previous stage. It is then of interest to determine conditions that guarantee strict error reduction at each stage forjrarious transport problems. Specifically, if Φ(x) is the true transport solution and \( \tilde \Phi ^{n - 1} (x) \) and \( \tilde \Phi ^n (x) \) are the estimated solutions from the (n-l)st and nth stages, respectively, we demonstrate the existence of a number λ, 0 < λ < 1, which is independent of the stage number n, such that

Geometric Convergence of Adaptive Monte Carlo Algorithms for Radiative Transport Problems Based on Importance Sampling Methods

\left\| {{{\tilde \Phi}^n}(x)-\Phi (x)} \right\| \leqslant \lambda \left\| {{{\tilde \Phi}^{n-1}}(x)-\tilde \Phi (x)} \right\| + \in

Adaptive Monte Carlo Algorithms Applied to Heterogeneous Transport Problems

in a certain probabilistic sense, where e is an error term that tends to zero as both the number of terms representing the global solution and the number of random walks per stage tend to infinity. We will indicate how to find such a λ, which is defined in terms of the number of random walks per stage and the coefficients of the transport problem in a rather natural way.

A new proof of geometric convergence for the adaptive generalized weighted analog sampling (GWAS) method

Conventional wisdom in solving transport problems is to identify unbiased, low variance (or low variation) estimators to estimate unknown functionals of the solution by Monte Carlo (or quasi-Monte Carlo) algorithms. Our adaptive implementations using this approach have involved the iterative improvement of either an approximate solution obtained through correlated sampling (SCS) or of an approximate importance function (AIS) for the problem. Each of these methods has some drawbacks: for SCS, the (required) estimation of the residual creates various problems and for AIS, sampling from the complex expressions that result from the use of an importance function can be extremely costly. In both of these cases, substantial loss of precision may result. A new adaptive method — generalized weighted analog sampling (GWAS) - combines many of the best features of SCS (simple sampling functions) and AIS (rapid error reduction) and makes use of biased (but asymptotically unbiased) estimators in a very flexible and efficient algorithm. In this work we sketch the needed theory and present numerical results that confirm the potential of the new method, at least for some model transport problems.

Transport-constrained extensions of collision and track length estimators for solutions of radiative transport problems

Abstract Importance sampling is a very well-known variance-reducing technique used in Monte Carlo simulations of radiative transport. It involves a distortion of the physical (analog) transition probabilities with the goal of causing events of interest in the computation to occur more frequently than in the analog process. This distortion is then compensated by a corresponding alteration of the estimating random variable in order to remove any bias from the estimates of quantities of interest. In this paper, we construct several families of estimators based on importance sampling methods to solve general transport problems and prove that the adaptive application of each estimator produces geometric convergence of the approximate solution. We also present numerical results that illustrate important elements of the theory.